Vector-valued Forms
Let E → M be a vector bundle. An E-valued differential form of degree r is a section of the tensor product bundle E ⊗ ΛrT*M. The space of such forms is denoted by
An E-valued 0-form is just a section of the bundle E. That is,
In this notation a connection on E → M is a linear map
A connection may then be viewed as a generalization of the exterior derivative to vector bundle valued forms. In fact, given a connection ∇ on E there is a unique way to extend ∇ to a covariant exterior derivative or exterior covariant derivative
Unlike the ordinary exterior derivative one need not have (d∇)2 = 0. In fact, (d∇)2 is directly related to the curvature of the connection ∇ (see below).
Read more about this topic: Connection (vector Bundle)
Famous quotes containing the word forms:
“The time will come when the evil forms we have known can no more be organized. Man’s culture can spare nothing, wants all material. He is to convert all impediments into instruments, all enemies into power.”
—Ralph Waldo Emerson (1803–1882)